期刊
ACM TRANSACTIONS ON GRAPHICS
卷 38, 期 1, 页码 -出版社
ASSOC COMPUTING MACHINERY
DOI: 10.1145/3152156
关键词
Shape analysis; geometry processing; Steklov eigenvalue problem; Dirichlet-to-Neumann operator
资金
- MIT-Skoltech Seed Fund grant (Boundary Element Methods for Shape Analysis)
- Skoltech-MIT Next Generation Program
- MIT Research Support Committee
- Army Research Office [W911NF-12-R-0011]
- National Science Foundation [IIS-1838071]
- Amazon Research Award
- NSERC
- FRQNT
- Canada Research Chairs program
- Weston Visiting Professorship program of the Weizmann Institute of Science
- European Research Council (ERC) [714776]
- Israel Science Foundation [504/16]
- MIT Grier Presidential Fellowship
- Siebel Scholarship
- European Research Council (ERC) [714776] Funding Source: European Research Council (ERC)
We propose using the Dirichlet-to-Neumann operator as an extrinsic alternative to the Laplacian for spectral geometry processing and shape analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator, cannot capture the spatial embedding of a shape up to rigid motion, and many previous extrinsic methods lack theoretical justification. Instead, we consider the Steklov eigenvalue problem, computing the spectrum of the Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable property of this operator is that it completely encodes volumetric geometry. We use the boundary element method (BEM) to discretize the operator, accelerated by hierarchical numerical schemes and preconditioning; this pipeline allows us to solve eigenvalue and linear problems on large-scale meshes despite the density of the Dirichlet-to-Neumann discretization. We further demonstrate that our operators naturally fit into existing frameworks for geometry processing, making a shift from intrinsic to extrinsic geometry as simple as substituting the Laplace Beltrami operator with the Dirichlet-to-Neumann operator.
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